Dividing S^3 into two separate pieces.

[Spinnor]

Take two three-dimensional balls, superimpose them, and identify their outer surfaces. This is a "standard" representation of S^3?

Clearly the common outer surface divides the two halves of S^3 into two separate volumes?

In a similar manner can the surface of a torus divide S^3 into two separate volumes? and the ratio of those two volumes being any number we please?

Thank you for any help.

 

[Spinnor]

I wrote:

...In a similar manner can the surface of a torus divide S^3 into two separate volumes? and the ratio of those two volumes being any number we please?...

This seems obvious, let us sit in S^3 with a garden hose. Connect the ends of the hose together. Letting the garden hose represent a torus we have separated the space S^3 into two volumes, inside the hose and outside the hose. Now let the garden hose grow in length while minimizing the total curvature of the hose, so that it "straightens out". At some length the hose will now acquire some minimum curvature, it has straightened out in S^3. At any point on the surface of the garden hose there will be two directions which are geodesics. One around the small radius of the hose and the other direction which takes us around the long radius of the hose.

If I allow the small radius of the hose to grow I think the large radius will decrease and at some point with radii equal we will have a torus in S^3 which separates S^3 into two equal sized volumes with equal radii?